The disconnect between control practice and control theory stems not only from differing orientations in theoretical research and practice, but also significantly from a lack of a common language for communication—namely, a lack of mathematical descriptions (i.e., models) of real-world controlled systems. As shown in Figure 1(a), on the one hand, control theorists directly start from mathematical formulas (usually differential equations), using mathematical derivation and proof to provide mathematical system properties (such as stability regions, observability, and controllability) and theoretical control laws. They lack attention to practical factors such as how these formulas are derived, the extent to which they describe real-world systems, how each mathematical variable in the formula relates to physical quantities in the real-world controlled system, whether these mathematical variables can be accurately obtained in real time, and the computational resources required to calculate these formulas. On the other hand, engineers face specific controlled systems (completing specific tasks/processes), needing to handle process logic, signal acquisition/processing, hardware configuration, and software programming implementation, and facing the pressure of on-site debugging. Naturally, they have neither the time nor the energy to deeply consider how their controlled systems can be described mathematically or what category of problems they are essentially categorized into.
The lack of a common language between control practice and control theory has resulted in several problems. On the one hand, pressing control challenges in reality heavily rely on PID tuning experience and manual intervention, hindering effective automation, let alone digitization and intelligentization. On the other hand, many mathematically rigorous and elegant control theories cannot be effectively adapted to practical situations, thus preventing their effective implementation and promotion in industry. Modeling, as an effective method for establishing this communication channel, faces a situation of being rendered speechless, which the author deeply regrets.
Even for data-driven control algorithms that aim to move away from reliance on mechanistic models, establishing a suitable mathematical description (not necessarily precise, but enough to roughly describe the system's main dynamic characteristics) is still necessary for practical application in real-world controlled systems. This aids in the design and debugging of the control algorithm, avoiding unnecessary trial-and-error time and costs. For example, for PID controllers (essentially data-driven controllers), if the system is a first-order inertial element (such as the current loop or speed loop in motor control), a PI controller is usually sufficient; however, if the object is a second-order element (such as a position control system), it is usually necessary to consider using derivative functions to improve phase lag and increase the system damping coefficient.
A correct understanding and serious consideration of the role of modeling in control algorithms is particularly important (Figure 1b) and urgent, especially today, with the upgrading and transformation of the manufacturing industry and the emphasis on original independent innovation. "A good control algorithm should be a perfect combination of strong physical concepts and mathematical tools, that is, based on a deep understanding of the problems in the real physical world, finding appropriate mathematical tools to describe and solve them. Specifically, in the design of control algorithms, system dynamics analysis and modeling, as tools connecting the physical world and the mathematical world, are particularly important."
Figure 1. Relationship between control algorithms, control theory, and control practice.
What is a model?
First, it should be clear that models exist for quantitative analysis and accurate description of the controlled system, thereby designing suitable control algorithms. Here, "model" refers to the mathematical description of the main problems of the real-world controlled system. In practice, many people believe that mathematical models are unnecessary and complicate the problem; they think it's sufficient to simply debug the machine on-site based on PID parameter tuning experience, rather than involving a bunch of mathematical formulas. However, even for on-site PID debugging, appropriate models and analysis based on them are still necessary.
• You need to know how the controllable parts (control inputs) act on the system and affect the system output, whether they have a positive or negative effect, whether they take effect immediately, after a period of time, or gradually, as shown in Figure 2, which shows the step curves of several common systems (normalized for easy comparison).
• You need to know the working process/conditions of the machine you are dealing with, which parameters/conditions will change under these conditions, the magnitude and speed of the changes, and how they affect your system;
• You need to know your control objectives, such as accuracy, dynamic response/bandwidth, and allowable overshoot.
All of the above requires some quantitative description and theoretical analysis to help you find the direction for debugging. For example, for the typical step response curves shown in Figure 2, the transfer function model can be used to analyze and determine the system type (such as time constant/response speed, whether there is a delay, the ratio of delay to time constant, whether it is underdamped, whether it is a minimum phase system), etc. Determining this information helps in selecting a suitable PID controller design strategy and is the foundation of PID auto-tuning. In fact, depending on the type of problem being described, the model can take many forms. If it is a dynamic process, it can be described using a frequency domain transfer function or a time domain state-space function, and it can be described in the continuous domain or the discrete domain; if it is a static nonlinear relationship, it can be described using polynomials or special functions, or even fuzzy logic, neural networks, etc.; or a combination of both to form the Hammerstein-Wiener model.
Figure 2. Typical step response curves and corresponding transfer function moduli (delay elements are commonly found in process control systems).
How to model it?
In the context of control algorithm design, modeling is guided by the design/implementation of the control algorithm. It does not mean modeling the entire system, but rather modeling the parts that are most relevant to control performance and can represent the main problems of the current controlled system.
For example, in robot control systems, for low-speed applications with low precision requirements, it's unnecessary to involve dynamic modeling; knowing the forward and inverse kinematic models is sufficient. However, for high-speed, high-precision applications, considering dynamic models, delving into joint flexibility, and even accurately modeling friction forces becomes essential, as joint flexibility and friction forces determine the robot's control performance in these situations.
Therefore, the first principle of modeling is to determine the purpose of modeling and the scope of the modeling (the main problem being described). Based on this, the second principle is that the model structure should match the structural characteristics of the object being described; that is, the model should be able to intuitively and explicitly describe the problem, ideally allowing direct analysis of the problem's essence from the mathematical expressions within the model. For controlled systems, various types of problems (such as the characteristics of the controlled object, feedback noise, external disturbances, etc.) can dominate the performance of the control system. Different model forms and structures are necessary to describe different types of problems. For example, if the controlled object is a time-invariant linear dynamic system, and only the relationship between input and output is of concern, then a frequency domain transfer function based on the Laplace transform is appropriate; if the observation or feedback of other related state variables is also involved, then a state-space representation can be considered; if the controlled object is a nonlinear dynamic system, then time-domain differential equations based on state variables are generally considered.
• If the controlled object exhibits parameter variations, requiring parameter identification, the analytical parameter model can be written in different forms based on information such as the rate/pattern of parameter changes, whether they are bounded, and their relationship with input/output data. For example, in parameter identification within a robot control system, it is necessary to combine some parameters to be identified into a parameter vector. To avoid using acceleration information, special modifications to the analytical parameter model are required (see Slotine and Li's Approach).
• In cases involving high-frequency unmodeled dynamics and uncertainties, if the structure of this unknown information or how it affects the system is known, it should be expressed mathematically (additive/multiplicative uncertainty, whether it is bounded, whether they are independent, etc.).
• If the mechanism of the controlled object cannot be accurately known, and an accurate mechanism model cannot be established through theoretical derivation starting from first principles, and only a data model can be established, then the input-output relationship should be explicitly represented. For example, how many previous input times are associated with the output at the current moment, and through what channels does the disturbance/noise enter the system (refer to the relevant practices of model-free adaptive control).
• If describing external disturbances, such as nonlinear friction forces or torque fluctuations, polynomial fitting or harmonic superposition can be considered; if it is also necessary to describe the dynamics of external disturbances or their relationship with the state variables of the control system, state-space representation can be considered to extend the original dynamic system state variables.
• If the feedback noise is caused by power frequency interference/inverter interference, it can be described by harmonic superposition. If it is random noise, it can be described by white noise, etc.
Properly describing the above issues helps to analyze the problems from their essence, more accurately match existing control algorithms (for example, the Kalman filter commonly used in control algorithms assumes that the noise is white noise, which may not be suitable for periodic noise interference and needs to be modified or other filtering methods should be selected), and maximize the organization of known information, thereby laying the foundation for designing efficient control algorithms (the more fully the control algorithm utilizes known information, the better the control effect achieved. For example, for adaptive control algorithms, knowing the upper and lower bounds of parameters can ensure that the parameters are bounded and improve transient performance; for robust control, if the structural information of the uncertainty is known, the conservatism of the control algorithm can be reduced).
Based on selecting a suitable model structure/form, the third principle of modeling is that the model should balance accuracy and real-time computational performance. In the context of control algorithm design, model information is needed in different modules of the control algorithm, as shown in Figure 3. Model information (the model generally needs to undergo some form of transformation before it can be used in the control system; therefore, "needs" in the figure indicates implicit/indirect use of model information) can be used in the outer loop forward channel composed of input shaping/trajectory planning and feedforward compensation, or in the feedback controller (such as model prediction, adaptive, robust/sliding mode control), or in the inner loop observer channel (such as state/disturbance estimation and parameter identification). These modules more or less require real-time computation of the model. If the model is too complex, although it can describe the problem relatively accurately, it also increases the real-time computation load (requiring a higher-performance controller or a relatively long computation cycle), and also leads to a higher controller order, increasing the difficulty of control system analysis and design. Therefore, a trade-off should be made between model accuracy and real-time computation while meeting control performance requirements.
Figure 3. Model information used for control algorithm design
Finally, the fourth principle of modeling is that the model must undergo some form of verification/actual testing before it can be used in a control system to ensure its correctness and accuracy (correctness requires the model to reflect the main changing patterns of the problem, while accuracy requires that the error between the model and reality be relatively small).
Summarize
Establishing a mathematical model of the controlled system, analyzing its essential characteristics, and developing physical intuition are all crucial in the design and debugging stages of control algorithms.
• In the control algorithm design phase, based on a thorough analysis of the controlled system, modeling not only enables a scientific analysis of the real problems faced but also provides the key to entering the edifice of control theory. This allows for the selection of a generally correct direction and appropriate control algorithm type at the early stages of a project, followed by the selection of suitable control methods and their necessary modifications, combinations, and even innovations and advancements. This abstraction from the physical world to the mathematical world is necessary and crucial for truly understanding things in their essence and achieving original innovation.
• During the algorithm debugging phase, in-depth analysis of the model and physical intuition help engineers quickly identify problems encountered during debugging and analyze possible causes. This provides the correct debugging direction or algorithm improvement methods, avoiding unnecessary trial and error and shortening the project cycle. This return from the mathematical world to the physical world is also necessary, as it helps theoretical innovations truly take root and be continuously improved in practice.
[Author Biography] Li Lei holds a PhD in Mechanical Engineering from Zhejiang University and was a visiting scholar at Georgia Institute of Technology (2016-2017). He currently works on the research and development of automation control algorithms. During his doctoral studies, he published several articles in journals such as IEEE TMech and TIE. He currently serves as a reviewer for international mechanical engineering journals such as TMech and IJIRA (International Journal of Intelligent Robotics and Applications).
